The QFI matrix plays a central role in quantum metrology and parameter estimation.
It also appears in the generalizations of the uncertainty relation that allow for the formulation of the time-energy uncertainty principle (time having no associated Hermitian operator and thus being a parameter that must be inferred by other measurements) [1]}, [2]}. Most importantly for our purposes, the mean trace of the QFI matrix quantifies the degree to which the state \(\psi \)  has genuine multipartite entanglement (GME)  [3]}. This connection is especially interesting as it holds even at finite temperatures [4]} (we discuss the details in Sec. REF ). Entries of the QFI matrix have now been studied in a wide range of models [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}. These studies have emphasized either parameterizations that are linear in the spin degrees of freedom, or in the case of  [12]}, parameterizations using non-local operators that exhibit super extensive scaling in topologically non trivial phases.
The full QFI matrix that would correspond to these parameterizations has been relatively less explored.
